He speaks English. But I don’t.
He doesn’t speak English. Nor do I.
I found the answer. But Alice didn’t.
I couldn’t find the answer. Neither did Alice.
He can cook. But his wife can’t.
He can’t cook. Nor can his wife.
I can play chess. But my sister can’t.
I can’t play chess. Neither can my sister.

Affirmative additions to negative remarks are made with ‘but + subject + auxiliary’.

He doesn’t know how to cook. But I do.
I didn’t see the film. But my sister did.
He can’t play chess. But I can.
She wasn’t late. But I was.